16,220 research outputs found
Polynomial tuning of multiparametric combinatorial samplers
Boltzmann samplers and the recursive method are prominent algorithmic
frameworks for the approximate-size and exact-size random generation of large
combinatorial structures, such as maps, tilings, RNA sequences or various
tree-like structures. In their multiparametric variants, these samplers allow
to control the profile of expected values corresponding to multiple
combinatorial parameters. One can control, for instance, the number of leaves,
profile of node degrees in trees or the number of certain subpatterns in
strings. However, such a flexible control requires an additional non-trivial
tuning procedure. In this paper, we propose an efficient polynomial-time, with
respect to the number of tuned parameters, tuning algorithm based on convex
optimisation techniques. Finally, we illustrate the efficiency of our approach
using several applications of rational, algebraic and P\'olya structures
including polyomino tilings with prescribed tile frequencies, planar trees with
a given specific node degree distribution, and weighted partitions.Comment: Extended abstract, accepted to ANALCO2018. 20 pages, 6 figures,
colours. Implementation and examples are available at [1]
https://github.com/maciej-bendkowski/boltzmann-brain [2]
https://github.com/maciej-bendkowski/multiparametric-combinatorial-sampler
Automated Circuit Approximation Method Driven by Data Distribution
We propose an application-tailored data-driven fully automated method for
functional approximation of combinational circuits. We demonstrate how an
application-level error metric such as the classification accuracy can be
translated to a component-level error metric needed for an efficient and fast
search in the space of approximate low-level components that are used in the
application. This is possible by employing a weighted mean error distance
(WMED) metric for steering the circuit approximation process which is conducted
by means of genetic programming. WMED introduces a set of weights (calculated
from the data distribution measured on a selected signal in a given
application) determining the importance of each input vector for the
approximation process. The method is evaluated using synthetic benchmarks and
application-specific approximate MAC (multiply-and-accumulate) units that are
designed to provide the best trade-offs between the classification accuracy and
power consumption of two image classifiers based on neural networks.Comment: Accepted for publication at Design, Automation and Test in Europe
(DATE 2019). Florence, Ital
From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
The next few years will be exciting as prototype universal quantum processors
emerge, enabling implementation of a wider variety of algorithms. Of particular
interest are quantum heuristics, which require experimentation on quantum
hardware for their evaluation, and which have the potential to significantly
expand the breadth of quantum computing applications. A leading candidate is
Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates
between applying a cost-function-based Hamiltonian and a mixing Hamiltonian.
Here, we extend this framework to allow alternation between more general
families of operators. The essence of this extension, the Quantum Alternating
Operator Ansatz, is the consideration of general parametrized families of
unitaries rather than only those corresponding to the time-evolution under a
fixed local Hamiltonian for a time specified by the parameter. This ansatz
supports the representation of a larger, and potentially more useful, set of
states than the original formulation, with potential long-term impact on a
broad array of application areas. For cases that call for mixing only within a
desired subspace, refocusing on unitaries rather than Hamiltonians enables more
efficiently implementable mixers than was possible in the original framework.
Such mixers are particularly useful for optimization problems with hard
constraints that must always be satisfied, defining a feasible subspace, and
soft constraints whose violation we wish to minimize. More efficient
implementation enables earlier experimental exploration of an alternating
operator approach to a wide variety of approximate optimization, exact
optimization, and sampling problems. Here, we introduce the Quantum Alternating
Operator Ansatz, lay out design criteria for mixing operators, detail mappings
for eight problems, and provide brief descriptions of mappings for diverse
problems.Comment: 51 pages, 2 figures. Revised to match journal pape
Space-time paraproducts for paracontrolled calculus, 3d-PAM and multiplicative Burgers equations
We sharpen in this work the tools of paracontrolled calculus in order to
provide a complete analysis of the parabolic Anderson model equation and
Burgers system with multiplicative noise, in a -dimensional Riemannian
setting, in either bounded or unbounded domains. With that aim in mind, we
introduce a pair of intertwined space-time paraproducts on parabolic H\"older
spaces, with good continuity, that happens to be pivotal and provides one of
the building blocks of higher order paracontrolled calculus.Comment: v3, 56 pages. Different points about renormalisation matters have
been clarified. Typos correcte
QCD sum rules at finite density in the large-N_c limit: The coupling of the rho-nucleon system to the D_{13}(1520)
QCD sum rules are studied for the vector-isovector current at finite baryon
density in the limit of large number of colors N_c. For the condensate side it
is shown that in this limit the four-quark condensate factorizes also for the
finite density case. At the hadronic side the medium dependence is expressed in
terms of the current-nucleon forward scattering amplitude. Generalizing vector
meson dominance we allow for a direct coupling of the current to the nucleon as
well as a coupling via the rho meson. We discuss the N_c dependence of (a)
modifications of the pion cloud of the rho meson, (b) mixing with other mesons
(in particular a_1 and omega) and (c) resonance-hole excitations R N^{-1}. We
show that only the last effect survives in the large-N_c limit. Saturating the
sum rules with a simple hadronic ansatz which allows for the excitation of the
D_{13}(1520) we determine the coupling of the latter to the rho-nucleon and the
photon-nucleon system. These couplings are hard to determine from vacuum
physics alone.Comment: 13 pages, 2 figure
Approximate Degree, Secret Sharing, and Concentration Phenomena
The epsilon-approximate degree deg~_epsilon(f) of a Boolean function f is the least degree of a real-valued polynomial that approximates f pointwise to within epsilon. A sound and complete certificate for approximate degree being at least k is a pair of probability distributions, also known as a dual polynomial, that are perfectly k-wise indistinguishable, but are distinguishable by f with advantage 1 - epsilon. Our contributions are:
- We give a simple, explicit new construction of a dual polynomial for the AND function on n bits, certifying that its epsilon-approximate degree is Omega (sqrt{n log 1/epsilon}). This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the 1/3-approximate degree of any (possibly unbalanced) read-once DNF is Omega(sqrt{n}). It draws a novel connection between the approximate degree of AND and anti-concentration of the Binomial distribution.
- We show that any pair of symmetric distributions on n-bit strings that are perfectly k-wise indistinguishable are also statistically K-wise indistinguishable with at most K^{3/2} * exp (-Omega (k^2/K)) error for all k < K <= n/64. This bound is essentially tight, and implies that any symmetric function f is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-K coalitions with statistical error K^{3/2} * exp (-Omega (deg~_{1/3}(f)^2/K)) for all values of K up to n/64 simultaneously. Previous secret sharing schemes required that K be determined in advance, and only worked for f=AND. Our analysis draws another new connection between approximate degree and concentration phenomena.
As a corollary of this result, we show that for any d deg~_{1/3}(f). These upper and lower bounds were also previously only known in the case f=AND
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